cp's OEIS Frontend

A276454 a(n) = A276452(n) + A276451(n) + A276449(n).

%I A276454 #32 Mar 26 2023 10:30:20
%S A276454 1,2,22,464,13302,487152,21475652,1106550392,65221981530,
%T A276454 4327577893800,319187492622012,25904823495240144,2294089575287710984,
%U A276454 220132629099295901408,22751391952803426496488,2519687900505935894639088,297684761086123702744203918
%N A276454 a(n) = A276452(n) + A276451(n) + A276449(n).
%C A276454 For a definition and examples of this problem see the comment section of A276449.
%C A276454 The present sequence {a(n)} gives the number of all orbits under C_4 of 2-colored n X n square grids with n squares of one color.
%C A276454 See A054772(n, k) for the table of these total C_4 orbit numbers for 2-colored grids with any number k from {0,1,...,n^2} of squares of one color. - _Wolfdieter Lang_, Oct 02 2016
%H A276454 Hong-Chang Wang, <a href="/A276454/b276454.txt">Table of n, a(n) for n = 1..70</a>
%F A276454 a(n) = A276452(n) + A276451(n) + A276449(n) for n = 1, 2, 3, ...,
%F A276454 A014062(n) = A276452(n)*4 + A276451(n)*2 + A276449(n).
%F A276454 a(n) = A054772(n, 2), n >= 1. - _Wolfdieter Lang_, Oct 02 2016
%e A276454 For n = 4 there are A276449(4) = 4 1-orbits, represented by
%e A276454    + o o +   o + o o   o o + o   o o o o
%e A276454    o o o o   o o o +   + o o o   o + + o
%e A276454    o o o o   + o o o   o o o +   o + + o
%e A276454    + o o +   o o + o   o + o o   o o o o  .
%e A276454 A276451(4) = 12 2-orbits: one of them is
%e A276454    + o + o   o o o +
%e A276454    o o o o   + o o o
%e A276454    o o o o   o o o +
%e A276454    o + o +   + o o o  ,
%e A276454 and one can take the first one as representative.
%e A276454 A276452(4) = 448 4-orbits: one of them is represented by
%e A276454    + + + +
%e A276454    o o o o
%e A276454    o o o o
%e A276454    o o o o .
%e A276454 The complete orbit structure for n=4 is 1^4 2^12 4^448, see A276449(4) = 4, A276451(4) = 12, A276452(4) = 448.
%e A276454 a(4) = 448 + 12 + 4 = 464.
%e A276454 A014062(4) = 448*4 + 12*2 + 4*1 = 1820.
%t A276454 f[n_] := If[MemberQ[{2, 3}, #], 0, Function[i, Binomial[(2 i) (2 i + #), i]]@ Floor[n/4]] &@ Mod[n, 4];g[n_] := (Function[j, Binomial[2 j (j + Boole@ OddQ@ n), j]]@ Floor[n/2] - f@ n)/2; Table[(Binomial[n^2, n] - 2 g@ n - f@ n)/4 + (Function[j, Binomial[2 j (j + Boole@ OddQ@ n), j]]@ Floor[n/2] - f@ n)/2 + f@ n, {n, 17}] (* _Michael De Vlieger_, Sep 12 2016 *)
%o A276454 (Python)
%o A276454 from math import comb as binomial
%o A276454 for j in range(1, 20):
%o A276454     t = binomial(j * j, j)
%o A276454     i = j // 2
%o A276454     if j % 2 == 0:
%o A276454         d = binomial(2 * i * i, i)
%o A276454     else:
%o A276454         d = binomial(2 * i * (i + 1), i)
%o A276454     a = (t - d) // 4
%o A276454     if j % 4 == 0:
%o A276454         c = binomial((j * j // 4), (j // 4))
%o A276454     elif j % 4 == 1:
%o A276454         c = binomial(((j - 1) // 2) * ((j - 1) // 2 + 1), ((j - 1) // 4))
%o A276454     else:
%o A276454         c = 0
%o A276454     b = (d - c) // 2
%o A276454     print(str(j) + " " + str(a + b + c))
%Y A276454 Cf. A014062, A054772, A276451, A276452, A276454.
%K A276454 nonn,easy
%O A276454 1,2
%A A276454 _Jason Y.S. Chiu_, _Hong-Chang Wang_, _Chiang, Tung-Ying_, Sep 03 2016
%E A276454 Edited by _Wolfdieter Lang_, Oct 02 2016